Chapter 7 – Limits, Continuity and Differentiability (JEE)
1. Introduction
This chapter forms the **foundation of calculus**.
Differentiation, integration, and applications of calculus all depend on:
- Limits
- Continuity
- Differentiability
2. Concept of a Limit
The limit of a function describes the value that a function approaches
as the variable gets closer to a given number.
$\displaystyle \lim_{x \to a} f(x) = L$
Limit depends on values **near** the point, not necessarily at the point.
3. Left Hand Limit (LHL)
$\displaystyle \lim_{x \to a^-} f(x)$
Value approached by the function when $x$ approaches $a$ from the left.
4. Right Hand Limit (RHL)
$\displaystyle \lim_{x \to a^+} f(x)$
Value approached by the function when $x$ approaches $a$ from the right.
5. Existence of Limit
A limit exists at $x=a$ if and only if:
LHL = RHL
If LHL ≠ RHL, the limit does NOT exist.
6. Standard Limits (Must Memorize)
$\displaystyle \lim_{x\to0} \frac{\sin x}{x} = 1$
$\displaystyle \lim_{x\to0} \frac{1-\cos x}{x^2} = \frac12$
$\displaystyle \lim_{x\to0} \frac{e^x - 1}{x} = 1$
$\displaystyle \lim_{x\to0} \frac{\ln(1+x)}{x} = 1$
7. Limits Using Algebraic Simplification
Methods used:
- Factorization
- Rationalization
- Cancellation
Never substitute directly when you get $\frac{0}{0}$.
8. Limits Using Trigonometric Identities
Use standard trigonometric limits and identities to simplify expressions.
9. Limits at Infinity
$\displaystyle \lim_{x\to\infty} \frac{1}{x} = 0$
Compare highest powers of $x$ in numerator and denominator.
10. Continuity of a Function
A function $f(x)$ is continuous at $x=a$ if:
- $f(a)$ is defined
- $\lim_{x\to a} f(x)$ exists
- $\lim_{x\to a} f(x) = f(a)$
11. Continuity of Polynomial and Trigonometric Functions
All polynomial, exponential, logarithmic and trigonometric functions
are continuous in their domains.
12. Continuity of Piecewise Functions
To check continuity:
- Find LHL
- Find RHL
- Equate them with $f(a)$
13. Types of Discontinuity
| Type | Description |
|---|---|
| Removable | Hole in the graph |
| Jump | LHL ≠ RHL |
| Infinite | Function becomes infinite |
14. Differentiability
A function is differentiable at a point if its derivative exists at that point.
$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$
15. Relationship Between Continuity and Differentiability
Differentiability ⇒ Continuity
Continuity ⇏ Differentiability
16. Differentiability of Standard Functions
All polynomial, trigonometric, exponential and logarithmic functions
are differentiable in their domains.
17. Non-Differentiable Points
Functions are NOT differentiable at:
- Sharp corners
- Cusps
- Vertical tangents
- Points of discontinuity
18. Differentiability of Piecewise Functions
To check differentiability at $x=a$:
- Find Left Hand Derivative (LHD)
- Find Right Hand Derivative (RHD)
- Check LHD = RHD
19. Modulus Function
$|x| = \begin{cases}
x, & x \ge 0 \\
-x, & x < 0
\end{cases}$
$|x|$ is continuous everywhere but NOT differentiable at $x=0$.
20. Greatest Integer Function
The greatest integer function $[x]$ is discontinuous at all integers.
21. Common JEE Traps
- Assuming continuity implies differentiability
- Forgetting LHD and RHD check
- Wrong use of standard limits
- Ignoring domain of function
22. Final Revision Checklist
You have mastered this chapter if you can:
- Evaluate limits confidently
- Apply standard limits instantly
- Check continuity of any function
- Check differentiability of piecewise functions
- Identify non-differentiable points